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<h1><img
src="normal_voronoi_knauss_oesterle.jpg" alt="[voronoi]" align="middle"
height="100">qvoronoi -- Voronoi diagram</h1>

<p>The Voronoi diagram is the nearest-neighbor map for a set of
points.  Each region contains those points that are nearer
one input site than any other input site.  It has many useful properties and applications. See the
survey article by Aurenhammer [<a href="index.htm#aure91">'91</a>]
and the detailed introduction by O'Rourke [<a
href="index.htm#orou94">'94</a>]. The Voronoi diagram is the
dual of the <a href=qdelaun.htm>Delaunay triangulation</a>. </p>

<blockquote>
<dl>
    <dt><b>Example:</b> rbox 10 D3 | qvoronoi <a href="qh-opto.htm#s">s</a>
        <a href="qh-opto.htm#o">o</a> <a href="qh-optt.htm#TO">TO
        result</a></dt>
    <dd>Compute the 3-d Voronoi diagram of 10 random points. Write a
        summary to the console and the Voronoi vertices and
        regions to 'result'. The first vertex of the result
        indicates unbounded regions.</dd>

    <dt>&nbsp;</dt>
    <dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi
             <a href="qh-opto.htm#s">s</a>
        <a href="qh-opto.htm#o">o</a> <a href="qh-optt.htm#TO">TO
        result</a></dt>
    <dd>Compute the 2-d Voronoi diagram of a triangle and a small
        square. Write a
        summary to the console and Voronoi vertices and regions
        to 'result'. Report a single Voronoi vertex for
        cocircular input sites. The first vertex of the result
        indicates unbounded regions. The origin is the Voronoi
        vertex for the square.</dd>

    <dt>&nbsp;</dt>
    <dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi <a href="qh-optf.htm#Fv2">Fv</a>
        <a href="qh-optt.htm#TO">TO result</a></dt>
    <dd>Compute the 2-d Voronoi diagram of a triangle and a small
        square. Write a
        summary to the console and the Voronoi ridges to
        'result'. Each ridge is the perpendicular bisector of a
        pair of input sites. Vertex &quot;0&quot; indicates
        unbounded ridges. Vertex &quot;8&quot; is the Voronoi
        vertex for the square.</dd>

    <dt>&nbsp;</dt>
    <dt><b>Example:</b> rbox r y c G0.1 D2 | qvoronoi <a href="qh-optf.htm#Fi2">Fi</a></dt>
    <dd>Print the bounded, separating hyperplanes for the 2-d Voronoi diagram of a
              triangle and a small
        square.  Note the four hyperplanes (i.e., lines) for Voronoi vertex
                &quot;8&quot;.  It is at the origin.
                </dd>
</dl>
</blockquote>

<p>Qhull computes the Voronoi diagram via the <a href="qdelaun.htm">Delaunay
triangulation</a>. Each Voronoi
vertex is the circumcenter of a facet of the Delaunay
triangulation. Each Voronoi region corresponds to a vertex (i.e., input site) of the
Delaunay triangulation. </p>

<p>Qhull outputs the Voronoi vertices for each Voronoi region. With
option '<a href="qh-optf.htm#Fv2">Fv</a>',
it lists all ridges of the Voronoi diagram with the corresponding
pairs of input sites. With
options '<a href="qh-optf.htm#Fi2">Fi</a>' and '<a href="qh-optf.htm#Fo2">Fo</a>',
it lists the bounded and unbounded separating hyperplanes.
You can also output a single Voronoi region
for further processing [see <a href="#graphics">graphics</a>].</p>

<p>Use option '<a href="qh-optq.htm#Qz">Qz</a>' if the input is circular, cospherical, or
nearly so.  It improves precision by adding a point "at infinity," above the corresponding paraboloid.

<p>See <a href="http://www.qhull.org/html/qh-faq.htm#TOC">Qhull FAQ</a> (<a href="qh-faq.htm#TOC">local</a>) - Delaunay and
Voronoi diagram questions.</p>

<p>The 'qvonoroi' program is equivalent to
'<a href="qhull.htm#outputs">qhull v</a> <a href="qh-optq.htm#Qbb">Qbb</a>'.  It disables the following Qhull
<a href=qh-quick.htm#options>options</a>: <i>d n v Qbb QbB Qf Qg Qm
Qr Qv Qx Qz TR E V Fa FA FC FD FS Ft FV Gt Q0,etc</i>.

<p><b>Copyright &copy; 1995-2020 C.B. Barber</b></p>

<p>Voronoi image by KOOK Architecture, Silvan Oesterle and Michael Knauss.

<hr>
<h3><a href="#TOP">&#187;</a><a name="synopsis">qvoronoi synopsis</a></h3>

<pre>
qvoronoi -- compute the Voronoi diagram.
    input (stdin): dimension, number of points, point coordinates
    comments start with a non-numeric character

options (qvoronoi.htm):
    Qu   - compute furthest-site Voronoi diagram
    Tv   - verify result: structure, convexity, and in-circle test
    .    - concise list of all options
    -    - one-line description of all options
    -?   - this message
    -V   - version

output options (subset):
    Fi   - separating hyperplanes for bounded regions, 'Fo' for unbounded
    FN   - count and Voronoi vertices for each Voronoi region
    Fv   - Voronoi diagram as Voronoi vertices between adjacent input sites
    G    - Geomview output (2-d only)
    o    - OFF file format (dim, Voronoi vertices, and Voronoi regions)
    p    - Voronoi vertices
    QVn  - Voronoi vertices for input point n, -n if not
    s    - summary of results (default)
    TI file - input file, may be enclosed in single quotes
    TO file - output file, may be enclosed in single quotes

examples:
rbox c P0 D2 | qvoronoi s o         rbox c P0 D2 | qvoronoi Fi
rbox c P0 D2 | qvoronoi Fo          rbox c P0 D2 | qvoronoi Fv
rbox c P0 D2 | qvoronoi s Qu Fv     rbox c P0 D2 | qvoronoi Qu Fo
rbox c G1 d D2 | qvoronoi s p       rbox c P0 D2 | qvoronoi s Fv QV0
</pre>

<h3><a href="#TOP">&#187;</a><a name="input">qvoronoi input</a></h3>
<blockquote>
The input data on <tt>stdin</tt> consists of:
<ul>
    <li>dimension
    <li>number of points</li>
    <li>point coordinates</li>
</ul>

<p>Use I/O redirection (e.g., qvoronoi &lt; data.txt), a pipe (e.g., rbox 10 | qvoronoi),
or the '<a href="qh-optt.htm#TI">TI</a>' option (e.g., qvoronoi TI data.txt).

<p>For example, this is four cocircular points inside a square.  Their Voronoi
diagram has nine vertices and eight regions.  Notice the Voronoi vertex
at the origin, and the Voronoi vertices (on each axis) for the four
sides of the square.
<p>
<blockquote>
<tt>rbox s 4 W0 c G1 D2 &gt; data</tt>
<blockquote><pre>
2 RBOX s 4 W0 c D2
8
-0.4941988586954018 -0.07594397977563715
-0.06448037284989526 0.4958248496365813
0.4911154367094632 0.09383830681375946
-0.348353580869097 -0.3586778257652367
    -1     -1
    -1      1
     1     -1
     1      1
</pre></blockquote>

<p><tt>qvoronoi s p &lt; data</tt>
<blockquote><pre>

Voronoi diagram by the convex hull of 8 points in 3-d:

  Number of Voronoi regions: 8
  Number of Voronoi vertices: 9
  Number of non-simplicial Voronoi vertices: 1

Statistics for: RBOX s 4 W0 c D2 | QVORONOI s p

  Number of points processed: 8
  Number of hyperplanes created: 18
  Number of facets in hull: 10
  Number of distance tests for qhull: 33
  Number of merged facets: 2
  Number of distance tests for merging: 102
  CPU seconds to compute hull (after input): 0.094

2
9
4.217546450968612e-17 1.735507986399734
-8.402566836762659e-17 -1.364368854147395
0.3447488772716865 -0.6395484723719818
1.719446929853986 2.136555906154247e-17
0.4967882915039657 0.68662371396699
-1.729928876283549 1.343733067524222e-17
-0.8906163241424728 -0.4594150543829102
-0.6656840313875723 0.5003013793414868
-7.318364664277155e-19 -1.188217818408333e-16
</pre></blockquote>
</blockquote>

</blockquote>
<h3><a href="#TOP">&#187;</a> <a name="outputs">qvoronoi
outputs</a></h3>
<blockquote>

<p>These options control the output of Voronoi diagrams.</p>
<blockquote>

<dl compact>
    <dt>&nbsp;</dt>
    <dd><b>Voronoi vertices</b></dd>
    <dt><a href="qh-opto.htm#p">p</a></dt>
    <dd>print the coordinates of the Voronoi vertices.  The first line
        is the dimension.  The second line is the number of vertices.  Each
        remaining line is a Voronoi vertex.</dd>
    <dt><a href="qh-optf.htm#Fn">Fn</a></dt>
    <dd>list the neighboring Voronoi vertices for each Voronoi
        vertex.  The first line is the number of Voronoi vertices.  Each
                remaining line starts with the number of neighboring vertices.
                Negative vertices (e.g., <em>-1</em>) indicate vertices
        outside of the Voronoi diagram.
     In the circle-in-box example, the
                Voronoi vertex at the origin has four neighbors.</dd>
    <dt><a href="qh-optf.htm#FN">FN</a></dt>
    <dd>list the Voronoi vertices for each Voronoi region.  The first line is
        the number of Voronoi regions.  Each remaining line starts with the
        number of Voronoi vertices.  Negative indices (e.g., <em>-1</em>) indicate vertices
        outside of the Voronoi diagram.
        In the circle-in-box example, the four bounded regions are defined by four
        Voronoi vertices.</dd>

    <dt>&nbsp;</dt>
    <dt>&nbsp;</dt>
    <dd><b>Voronoi regions</b></dd>
    <dt><a href="qh-opto.htm#o">o</a></dt>
    <dd>print the Voronoi regions in OFF format.  The first line is the
        dimension.  The second line is the number of vertices, the number
        of input sites, and "1".  The third line represents the vertex-at-infinity.
        Its coordinates are "-10.101".  The next lines are the coordinates
        of the Voronoi vertices.  Each remaining line starts with the number
        of Voronoi vertices in a Voronoi region.  In 2-d, the vertices are
listed in adjacency order (unoriented). In 3-d and higher, the
vertices are listed in numeric order.  In the circle-in-square
        example, each bounded region includes the Voronoi vertex at
        the origin.  Lines consisting of <em>0</em> indicate
        coplanar input sites or '<a href="qh-optq.htm#Qz">Qz</a>'.  </dd>
    <dt><a href="qh-optf.htm#Fi2">Fi</a></dt>
    <dd>print separating hyperplanes for inner, bounded Voronoi
        regions.  The first number is the number of separating
                hyperplanes.  Each remaining line starts with <i>3+dim</i>.  The
                next two numbers are adjacent input sites.  The next <i>dim</i>
                numbers are the coefficients of the separating hyperplane.  The
                last number is its offset.  Use '<a href="qh-optt.htm#Tv">Tv</a>' to verify that the
hyperplanes are perpendicular bisectors. It will list relevant
statistics to stderr. </dd>
    <dt><a href="qh-optf.htm#Fo2">Fo</a></dt>
    <dd>print separating hyperplanes for outer, unbounded Voronoi
        regions.  The first number is the number of separating
                hyperplanes.  Each remaining line starts with <i>3+dim</i>.  The
                next two numbers are adjacent input sites on the convex hull.  The
                next <i>dim</i>
                numbers are the coefficients of the separating hyperplane.  The
                last number is its offset.</dd>
    <dt>&nbsp;</dt>
    <dt>&nbsp;</dt>
    <dd><b>Input sites</b></dd>
    <dt><a href="qh-optf.htm#Fv2">Fv</a></dt>
    <dd>list ridges of Voronoi vertices for pairs of input sites.  The
        first line is the number of ridges.  Each remaining line starts with
        two plus the number of Voronoi vertices in the ridge.  The next
        two numbers are two adjacent input sites.  The remaining numbers list
        the Voronoi vertices.  As with option 'o', a <em>0</em> indicates
        the vertex-at-infinity
        and an unbounded, separating hyperplane.
        The perpendicular bisector (separating hyperplane)
        of the input sites is a flat through these vertices.
        In the circle-in-square example, the ridge for each edge of the square
        is unbounded.</dd>
    <dt><a href="qh-optf.htm#Fc">Fc</a></dt>
    <dd>list coincident input sites for each Voronoi vertex.
        The first line is the number of vertices.  The remaining lines start with
        the number of coincident sites and deleted vertices.  Deleted vertices
        indicate highly degenerate input (see'<a href="qh-optf.htm#Fs">Fs</a>').
        A coincident site is assigned to one Voronoi
        vertex.  Do not use '<a href="qh-optq.htm#QJn">QJ</a>' with 'Fc'; the joggle will separate
        coincident sites.</dd>
    <dt><a href="qh-optf.htm#FP">FP</a></dt>
    <dd>print coincident input sites with distance to
        nearest site (i.e., vertex). The first line is the
        number of coincident sites.  Each remaining line starts with the point ID of
        an input site, followed by the point ID of a coincident point, its vertex, and distance.
        Includes deleted vertices which
        indicate highly degenerate input (see'<a href="qh-optf.htm#Fs">Fs</a>').
        Do not use '<a href="qh-optq.htm#QJn">QJ</a>' with 'FP'; the joggle will separate
        coincident sites.</dd>
    <dt>&nbsp;</dt>
    <dt>&nbsp;</dt>
    <dd><b>General</b></dd>
    <dt><a href="qh-opto.htm#s">s</a></dt>
    <dd>print summary of the Voronoi diagram. Use '<a
        href="qh-optf.htm#Fs">Fs</a>' for numeric data.</dd>
    <dt><a href="qh-opto.htm#i">i</a></dt>
    <dd>list input sites for each <a href=qdelaun.htm>Delaunay region</a>.  Use option '<a href="qh-optp.htm#Pp">Pp</a>'
        to avoid the warning.  The first line is the number of regions.  The
        remaining lines list the input sites for each region.  The regions are
        oriented.  In the circle-in-square example, the cocircular region has four
        edges.  In 3-d and higher, report cospherical sites by adding extra points.
        </dd>
    <dt><a href="qh-optg.htm#G">G</a></dt>
    <dd>Geomview output for 2-d Voronoi diagrams.</dd>
        </dl>
</blockquote>
</blockquote>
<h3><a href="#TOP">&#187;</a> <a name="controls">qvoronoi
controls</a></h3>
<blockquote>

<p>These options provide additional control:</p>
<blockquote>

<dl compact>
   <dt><a href="qh-optq.htm#Qu">Qu</a></dt>
    <dd>compute the <a href="qvoron_f.htm">furthest-site Voronoi diagram</a>.</dd>
    <dt><a href="qh-optq.htm#QRn">QRn</a></dt>
    <dd>randomly rotate the input with a random seed of n.  If n=0, the
        seed is the time.  If n=-1, use time for the random seed, but do
        not rotate the input.</dd>
    <dt><a href="qh-optq.htm#QVn">QVn</a></dt>
    <dd>select Voronoi vertices for input site <em>n</em> </dd>
    <dt><a href="qh-optq.htm#Qz">Qz</a></dt>
    <dd>add a point above the paraboloid to reduce precision
        errors. Use it for nearly cocircular/cospherical input
        (e.g., 'rbox c | qvoronoi Qz').</dd>
    <dt><a href="qh-optt.htm#Tv">Tv</a></dt>
    <dd>verify result</dd>
    <dt><a href="qh-optt.htm#TI">TI file</a></dt>
    <dd>input data from file.  The filename may not use spaces or quotes.</dd>
    <dt><a href="qh-optt.htm#TO">TO file</a></dt>
    <dd>output results to file.  Use single quotes if the filename
        contains spaces (e.g., <tt>TO 'file with spaces.txt'</tt></dd>
    <dt><a href="qh-optt.htm#TFn">TFn</a></dt>
    <dd>report progress after constructing <em>n</em> facets</dd>
    <dt><a href="qh-optp.htm#PDk">PDk:1</a></dt>
    <dd>include upper and lower facets in the output.  Set <em>k</em>
        to the last dimension (e.g., 'PD2:1' for 2-d inputs). </dd>
    <dt><a href="qh-opto.htm#f">f</a></dt>
    <dd>facet dump.  Print the data structure for each facet (i.e.,
        Voronoi vertex).</dd>
</dl>

</blockquote>
</blockquote>
<h3><a href="#TOP">&#187;</a> <a name="graphics">qvoronoi
graphics</a></h3>
<blockquote>

<p>In 2-d, Geomview output ('<a href="qh-optg.htm#G">G</a>')
displays a Voronoi diagram with extra edges to close the
unbounded Voronoi regions. To view the unbounded rays, enclose
the input points in a square.</p>

<p>You can also view <i>individual</i> Voronoi regions in 3-d. To
view the Voronoi region for site 3 in Geomview, execute</p>

<blockquote>
    <p>qvoronoi &lt;data <a href="qh-optq.htm#QVn">QV3</a> <a
    href="qh-opto.htm#p">p</a> | qconvex s G &gt;output</p>
</blockquote>

<p>The <tt>qvoronoi</tt> command returns the Voronoi vertices
for input site 3.  The <tt>qconvex</tt> command computes their convex hull.
This is the Voronoi region for input site 3.  Its
hyperplane normals (qconvex 'n') are the same as the separating hyperplanes
from options '<a href="qh-optf.htm#Fi">Fi</a>'
and '<a href="qh-optf.htm#Fo">Fo</a>' (up to roundoff error).

<p>See the <a href="qh-eg.htm#delaunay">Delaunay and Voronoi
examples</a> for 2-d and 3-d examples. Turn off normalization (on
Geomview's 'obscure' menu) when comparing the Voronoi diagram
with the corresponding Delaunay triangulation. </p>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="notes">qvoronoi
notes</a></h3>
<blockquote>

<p>You can simplify the Voronoi diagram by enclosing the input
sites in a large square or cube.  This is particularly recommended
for cocircular or cospherical input data.</p>

<p>See <a href="#graphics">Voronoi graphics</a> for computing
the convex hull of a Voronoi region. </p>

<p>Voronoi diagrams do not include facets that are
coplanar with the convex hull of the input sites.  A facet is
coplanar if the last coefficient of its normal is
nearly zero (see <a href="http://www.qhull.org/src/libqhull_r/user_r.h#ZEROdelaunay">qh_ZEROdelaunay</a>).

<p>Unbounded regions can be confusing. For example, '<tt>rbox c |
qvoronoi Qz o</tt>' produces the Voronoi regions for the vertices
of a cube centered at the origin. All regions are unbounded. The
output is </p>

<blockquote>
    <pre>3
2 9 1
-10.101 -10.101 -10.101
     0      0      0
2 0 1
2 0 1
2 0 1
2 0 1
2 0 1
2 0 1
2 0 1
2 0 1
0
</pre>
</blockquote>

<p>The first line is the dimension. The second line is the number
of vertices and the number of regions. There is one region per
input point plus a region for the point-at-infinity added by
option '<a href="qh-optq.htm#Qz">Qz</a>'. The next two lines
lists the Voronoi vertices. The first vertex is the infinity
vertex. It is indicate by the coordinates <em>-10.101</em>. The
second vertex is the origin. The next nine lines list the
regions. Each region lists two vertices -- the infinity vertex
and the origin. The last line is &quot;0&quot; because no region
is associated with the point-at-infinity. A &quot;0&quot; would
also be listed for nearly incident input sites. </p>

<p>To use option '<a href="qh-optf.htm#Fv">Fv</a>', add an
interior point. For example, </p>

<blockquote>
    <pre>
rbox c P0 | qvoronoi Fv
20
5 0 7 1 3 5
5 0 3 1 4 5
5 0 5 1 2 3
5 0 1 1 2 4
5 0 6 2 3 6
5 0 2 2 4 6
5 0 4 4 5 6
5 0 8 5 3 6
5 1 2 0 2 4
5 1 3 0 1 4
5 1 5 0 1 2
5 2 4 0 4 6
5 2 6 0 2 6
5 3 4 0 4 5
5 3 7 0 1 5
5 4 8 0 6 5
5 5 6 0 2 3
5 5 7 0 1 3
5 6 8 0 6 3
5 7 8 0 3 5
</pre>
</blockquote>

<p>The output consists of 20 ridges and each ridge lists a pair
of input sites and a triplet of Voronoi vertices. The first eight
ridges connect the origin ('P0'). The remainder list the edges of
the cube. Each edge generates an unbounded ray through the
midpoint. The corresponding separating planes ('Fo') follow each
pair of coordinate axes. </p>

<p>Options '<a href="qh-optq.htm#Qt">Qt</a>' (triangulated output)
and '<a href="qh-optq.htm#QJn">QJ</a>' (joggled input) are deprecated.  They may produce
unexpected results.   If you use these options, cocircular and cospherical input sites will
produce duplicate or nearly duplicate Voronoi vertices.  See also <a
href="qh-impre.htm#joggle">Merged facets or joggled input</a>. </p>

</blockquote>
<h3><a href="#TOP">&#187;</a><a name="conventions">qvoronoi conventions</a></h3>
<blockquote>

<p>The following terminology is used for Voronoi diagrams in
Qhull. The underlying structure is a Delaunay triangulation from
a convex hull in one higher dimension. Facets of the Delaunay
triangulation correspond to vertices of the Voronoi diagram.
Vertices of the Delaunay triangulation correspond to input sites.
They also correspond to regions of the Voronoi diagram. See <a
href="qconvex.htm#conventions">convex hull conventions</a>, <a
href="qdelaun.htm#conventions">Delaunay conventions</a>, and
<a href="index.htm#structure">Qhull's data structures</a>.</p>
<blockquote>

<ul>
    <li><em>input site</em> - a point in the input (one dimension
        lower than a point on the convex hull)</li>
    <li><em>point</em> - a point has <i>d+1</i> coordinates. The
        last coordinate is the sum of the squares of the input
        site's coordinates</li>
    <li><em>coplanar point</em> - a <em>nearly incident</em>
        input site</li>
    <li><em>vertex</em> - a point on the paraboloid. It
        corresponds to a unique input site. </li>
    <li><em>point-at-infinity</em> - a point added above the
        paraboloid by option '<a href="qh-optq.htm#Qz">Qz</a>'</li>
    <li><em>Delaunay facet</em> - a lower facet of the
        paraboloid. The last coefficient of its normal is
                clearly negative.</li>
    <li><em>Voronoi vertex</em> - the circumcenter of a Delaunay
        facet</li>
    <li><em>Voronoi region</em> - the Voronoi vertices for an
        input site. The region of Euclidean space nearest to an
        input site.</li>
    <li><em>Voronoi diagram</em> - the graph of the Voronoi
        regions. It includes the ridges (i.e., edges) between the
        regions.</li>
    <li><em>vertex-at-infinity</em> - the Voronoi vertex that
        indicates unbounded Voronoi regions in '<a
        href="qh-opto.htm#o">o</a>' output format. Its
        coordinates are <em>-10.101</em>.</li>
    <li><em>good facet</em> - a Voronoi vertex with optional
        restrictions by '<a href="qh-optq.htm#QVn">QVn</a>', etc.</li>
</ul>

</blockquote>
</blockquote>
<h3><a href="#TOP">&#187;</a><a name="options">qvoronoi options</a></h3>

<pre>
qvoronoi -- compute the Voronoi diagram
    http://www.qhull.org

input (stdin):
    first lines: dimension and number of points (or vice-versa).
    other lines: point coordinates, best if one point per line
    comments:    start with a non-numeric character

options:
    Qu   - compute furthest-site Voronoi diagram

Qhull control options:
    Qa   - allow input with fewer or more points than coordinates
    QRn  - random rotation (n=seed, n=0 time, n=-1 time/no rotate)
    Qs   - search all points for the initial simplex
    Qz   - add point-at-infinity to Voronoi diagram

Qhull extra options:
    QGn  - Voronoi vertices if visible from point n, -n if not
    QVn  - Voronoi vertices for input point n, -n if not
    Qw   - allow option warnings
    Q12  - allow wide facets and wide dupridge
    Q14  - merge pinched vertices that create a dupridge

T options:
    TFn  - report summary when n or more facets created
    TI file - input file, may be enclosed in single quotes
    TO file - output file, may be enclosed in single quotes
    Ts   - statistics
    Tv   - verify result: structure, convexity, and in-circle test
    Tz   - send all output to stdout

Trace options:
    T4   - trace at level n, 4=all, 5=mem/gauss, -1= events
    Ta   - annotate output with message codes
    TAn  - stop qhull after adding n vertices
     TCn - stop qhull after building cone for point n
     TVn - stop qhull after adding point n, -n for before
    Tc   - check frequently during execution
    Tf   - flush each qh_fprintf for debugging segfaults
    TPn  - turn on tracing when point n added to hull
     TMn - turn on tracing at merge n
     TWn - trace merge facets when width > n

Precision options:
    Cn   - radius of centrum (roundoff added).  Merge facets if non-convex
     An  - cosine of maximum angle.  Merge facets if cosine > n or non-convex
           C-0 roundoff, A-0.99/C-0.01 pre-merge, A0.99/C0.01 post-merge
    Rn   - randomly perturb computations by a factor of [1-n,1+n]
    Wn   - min facet width for non-coincident point (before roundoff)

Output formats (may be combined; if none, summary to stdout):
    p    - Voronoi vertices
    s    - summary to stderr
    f    - facet dump
    i    - Delaunay regions (use 'Pp' to avoid warning)
    o    - OFF format (dim, Voronoi vertices, and Voronoi regions)

More formats:
    Fc   - count plus coincident points (by Voronoi vertex)
    Fd   - use cdd format for input (homogeneous with offset first)
    FD   - use cdd format for output (offset first)
    FF   - facet dump without ridges
    Fi   - separating hyperplanes for bounded Voronoi regions
    FI   - ID for each Voronoi vertex
    Fm   - merge count for each Voronoi vertex (511 max)
    Fn   - count plus neighboring Voronoi vertices for each Voronoi vertex
    FN   - count and Voronoi vertices for each Voronoi region
    Fo   - separating hyperplanes for unbounded Voronoi regions
    FO   - options and precision constants
    FP   - nearest point and distance for each coincident point
    FQ   - command used for qvoronoi
    Fs   - summary: #int (8), dimension, #points, tot vertices, tot facets,
                    for output: #Voronoi regions, #Voronoi vertices,
                                #coincident points, #non-simplicial regions
                    #real (2), max outer plane and min vertex
    Fv   - Voronoi diagram as Voronoi vertices between adjacent input sites
    Fx   - extreme points of Delaunay triangulation (on convex hull)

Geomview output (2-d only)
    Ga   - all points as dots
     Gp  -  coplanar points and vertices as radii
     Gv  -  vertices as spheres
    Gc   - centrums
    GDn  - drop dimension n in 3-d and 4-d output
    Gh   - hyperplane intersections
    Gi   - inner planes only
     Gn  -  no planes
     Go  -  outer planes only
    Gr   - ridges

Print options:
    PAn  - keep n largest Voronoi vertices by 'area'
    Pdk:n - drop facet if normal[k] <= n (default 0.0)
    PDk:n - drop facet if normal[k] >= n
    PFn  - keep Voronoi vertices whose 'area' is at least n
    Pg   - print good Voronoi vertices (needs 'QGn' or 'QVn')
    PG   - print neighbors of good Voronoi vertices
    PMn  - keep n Voronoi vertices with most merges
    Po   - force output.  If error, output neighborhood of facet
    Pp   - do not report precision problems

    .    - list of all options
    -    - one line descriptions of all options
    -?   - help with examples
    -V   - version
</pre>

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